Analysis of the consistency of parity-odd nonbirefringent modified Maxwell theory

Analysis of the consistency of parity-odd nonbirefringent modified Maxwell theory

M. Schreck Institute for Theoretical Physics, Karlsruhe Institute of Technology (KIT),
76128 Karlsruhe, Germany
Abstract

There exist two deformations of standard electrodynamics that describe Lorentz symmetry violation in the photon sector: CPT-odd Maxwell–Chern–Simons theory and CPT-even modified Maxwell theory. In this article, we focus on the parity-odd nonbirefringent sector of modified Maxwell theory. It is coupled to a standard Dirac theory of massive spin-1/2 fermions resulting in a modified quantum electrodynamics (QED). This theory is discussed with respect to properties such as microcausality and unitarity, where it turns out that these hold.

Furthermore, a priori, the limit of the theory for vanishing Lorentz-violating parameters seems to be discontinuous. The modified photon polarization vectors are interweaved with preferred spacetime directions defined by the theory and one vector even has a longitudinal part. That structure remains in the limit mentioned. Since it is not clear, whether or not this behavior is a gauge artifact, the cross section for a physical process — modified Compton scattering — is calculated numerically. Despite the numerical instabilities occurring for scattering of unpolarized electrons off polarized photons in the second physical polarization state, it is shown that for Lorentz-violating parameters much smaller than one, the modified cross sections approach the standard QED results. Analytical investigations strengthen the numerical computations.

Hence, the theory proves to be consistent, at least with regard to the investigations performed. This leads to the interesting outcome of the modification being a well-defined parity-odd extension of QED.

Lorentz violation; parity violation; quantum electrodynamics; theory of quantized fields
pacs:
11.30.Cp, 11.30.Er, 12.20.-m, 03.70.+k

I Introduction

Modern quantum field theories are based on fundamental symmetries. This holds for quantum electrodynamics (QED) as well as for the standard model of elementary particle physics. Whenever physicists talk about symmetries they usually think of gauge invariance or the discrete symmetries charge conjugation C, parity P, and time reversal T. However, there is one symmetry that often takes a back seat: Lorentz invariance. This is not surprising, since until now there had been no convincing experimental evidence for a violation of Lorentz invariance.111 At the end of September 2011 this seemed to change with the publication of the result by the OPERA collaboration, which claimed to have discovered Lorentz violation in the neutrino sector OPERA:2011zb (). A large number of theoretical models emerged trying to explain the observed anomaly, for example by Fermi point splitting Klinkhamer:2011mf (), spontaneous symmetry breaking caused by the existence of a fermionic condensate Klinkhamer:2011iz (), or a multiple Lorentz group structure Schreck:2011ni (). However, the physics community remained sceptical and articles were published trying to explain the result by an error source that had not been taken into account Contaldi:2011 (); Besida:2011fi (); vanElburg:2011ze (). Unfortunately, at the 25th International Conference on Neutrino Physics and Astrophysics OPERA announced that their new measurement yields a deviation of the neutrino velocity from the speed of light, which is consistent with zero. Now again all laws of nature seem to obey Lorentz invariance.

However, a violation of other symmetries is part of the everyday life of any high-energy physicist. For example, violations of P and CP were measured long ago Wu:1957 (); Christenson:1964fg () and a broken electroweak gauge symmetry with massive , and bosons is an experimental fact. Why then should Lorentz symmetry and its violation not be of interest?

There exist good theoretical arguments for Lorentz invariance being a symmetry that is restored at low energies ChadhaNielsen1983 (). At the Planck length the topology of spacetime may be dynamical, which could lead to it having a foamy structure. The existence of such a spacetime foam Wheeler:1957mu (); Hawking:1979zw () may define a preferred reference frame — as is the case for water in a glass — and thus violate Lorentz invariance. Since a fundamental quantum theory of spacetime is still not known, we have to rely on well-established theories such as the standard model or special relativity for a description of Lorentz violation. By introducing new parameters that deform these theories it is possible to parameterize Lorentz violation on the basis of standard physics. One approach is to modify dispersion relations of particles. However, such a procedure is very ad hoc and it is not evident where the modification comes from. Therefore, a more elementary possibility is to parameterize modifications on the level of Lagrange densities. A collection of all Lorentz-violating deformations of the standard model that are gauge invariant is known as the Lorentz-violating extension of the standard model ColladayKostelecky1998 (). The minimal version of this extension relies on power-counting renormalizable terms, whereas the nonminimal version also includes operators of mass dimension (see e.g. the analyses performed in Kostelecky:2009zp (); Kostelecky:2011gq (); Cambiaso:2012vb ()).

The theoretical consistency of the standard model itself has been verified by investigations based on Lorentz-invariant quantum field theory that were performed over decades (see, for example, Ref. JordanPauli1928 ()). However, it is not entirely clear if a Lorentz-violating theory is consistent. Some results on certain sectors of the standard model extension already exist KosteleckyLehnert2000 (); AdamKlinkhamer2001 (); Liberati:2001sd (); Mavromatos:2009xg (); Casana-etal2009 (); Casana-etal2010 (); Klinkhamer:2010zs (); Klinkhamer:2011ez (), but there still remains a lot what we can learn about Lorentz-violating quantum field theories. Because of this it is very important to check Lorentz-violating deformations with respect to fundamental properties such as microcausality and unitarity. Furthermore, it is of significance whether the modified theory approaches the standard theory for arbitrarily small deformations. The purpose of this paper is to investigate these questions.

Especially in the case where Lorentz violation resides in the photon sector, it can lead to a variety of new effects, for example a birefringent vacuum ColladayKostelecky1998 (), new particle decays Beall:1970rw (); Coleman:1997xq (), and “aetherlike” deviations from special relativity, which are modulated with the rotation of the Earth around the Sun (e.g. Refs. Phillips:2000dr (); Bear:2000cd ()). From an experimental point of view, photons produce clean signals making the photon sector very important, in bounding Lorentz-violating parameters.

There exist two gauge-invariant and power-counting renormalizable deformations of the photon sector: Maxwell–Chern–Simons theory (MCS-theory) Carroll-etal1990 () and modified Maxwell theory ColladayKostelecky1998 (); KosteleckyMewes2002 (). Each Lagrangian contains additional terms besides the Maxwell term of standard electrodynamics. The consistency of the isotropic and one anisotropic sector of modified Maxwell theory was already shown in Klinkhamer:2010zs (). In this article a special sector, that violates parity and is supposed to show no birefringence, will be investigated.

The paper is organized as follows. In Sec. 2 modified Maxwell theory is presented and restricted to the parity-odd nonbirefringent case. Additionally, it is coupled to a standard Dirac theory of massive spin-1/2 fermions, which leads to a theory of modified QED. In Secs. 3 and 4, we review the nonstandard photon dispersion relations and the gauge propagator, which are determined from the field equations Casana-etal2009 (); Casana-etal2010 (). That completes the current status of research concerning this special sector of modified Maxwell theory. The successive parts of the article deal with the main issue, beginning with the deformed polarization vectors, which can also be obtained from the field equations. After setting up the building blocks we are ready to discuss unitarity in Sec. 6 and microcausality in Sec. 7. The subsequent two sections are devoted to the polarization vectors themselves. Since their form is rather uncommon — even when considering Lorentz-violating theories — we make comparisons with MCS-theory and other sectors of modified Maxwell theory. It will become evident that the polarization vectors have a property that distinguishes them from the polarization vectors of standard electrodynamics, even in the limit of vanishing Lorentz violation. To test, whether or not some residue of the deformation remains in this limit, in Sec. 9 we compute the cross section of the simplest tree-level process involving external modified photons that is also allowed by standard QED: Compton scattering. We conclude in the last section. Readers may skip Secs. 4 – 8 on first reading.

Ii Modified Maxwell theory

ii.1 Action and nonbirefringent Ansatz

In this article, we focus on modified Maxwell theory ChadhaNielsen1983 (); ColladayKostelecky1998 (); KosteleckyMewes2002 (). This particular Lorentz-violating theory is characterized by the action

 SmodMax = ∫R4d4xLmodMax(x), (2.1a) LmodMax(x) = −14ημρηνσFμν(x)Fρσ(x)−14κμνϱσFμν(x)Fϱσ(x), (2.1b)

which involves the field strength tensor of the gauge field . The fields are defined on Minkowski spacetime with global Cartesian coordinates and metric . The first term in Eq. (2.1b) represents the standard Maxwell term and the second corresponds to a modification of the standard theory of photons. The fixed background field selects preferred directions in spacetime and, therefore, breaks Lorentz invariance.

The second term in Eq. (2.1b) is expected to have the same symmetries as the first. These correspond to the symmetries of the Riemann curvature tensor, which reduces the number of independent parameters to 20. Furthermore, a vanishing double trace, , is imposed. A nonvanishing can be absorbed by a field redefinition ColladayKostelecky1998 () and does not contribute to physical observables. This additional condition leads to a remaining number of 19 independent parameters.

Modified Maxwell theory has two distinct parameter sectors that can be distinguished from each other by the property of birefringence. The first consists of 10 parameters and leads to birefringent photon modes at leading-order Lorentz violation. The second is made up of 9 parameters and shows no birefringence, at least to first order with respect to the parameters. Since the 10 birefringent parameters are bounded by experiment at the level Kostelecky:2001mb (), we will restrict our considerations to the nonbirefringent sector, which can be parameterized by the following Ansatz BaileyKostelecky2004 ():

 κμνϱσ=12(ημϱ˜κνσ−ημσ˜κνϱ−ηνϱ˜κμσ+ηνσ˜κμϱ), (2.2)

with a constant symmetric and traceless matrix . Here and in the following, natural units are used with , where corresponds to the maximal attainable velocity of the standard Dirac particles, whose action will be defined in Sec. II.3.

There exists a premetric formulation of classical electrodynamics, that is solely based on the concept of a manifold and does not need a metric. In this context a tensor density (electromagnetic field strength) and pseudotensor densities , (electromagnetic excitation and electric current) are introduced. Since the resulting field equations for these quantities are underdetermined, an additional relation between and has to be imposed, which is governed by the so-called constitutive four-tensor . Modified Maxwell theory emerges as one special case of this description, namely as the principal part of the constitutive tensor previously mentioned Hehl:2003 (); Itin:2009aa (). In Eq. (D.1.80) of the book Hehl:2003 () the nonbirefringent Ansatz of Eq. (2.2) can be found, as well. Section D.1.6 gives a motivation for it as the simplest — but not the most general — decomposition of the principal part of .

Furthermore, note that a special sector of CPT-even modified Maxwell theory arises as a contribution of the one-loop effective action of a CPT-odd deformation involving a spinor field and the photon field Gomes:2009ch ().

ii.2 Restriction to the parity-odd anisotropic case

The anisotropic case considered concerns the parity-odd sector of modified Maxwell theory (2.1) with the Ansatz from Eq. (2.2). This case is characterized by one purely timelike normalized four-vector and one purely spacelike four-vector containing three real parameters , , and  :

 ˜κμν = 12(ξμζν+ζμξν)−14ξλζλημν, (2.3a) (ξμ) = (1,0,0,0),(ζμ)≡(0,2ζ)=(0,2˜κ01,2˜κ02,2˜κ03), (2.3b) (˜κμν) = ⎛⎜ ⎜ ⎜ ⎜⎝0˜κ01˜κ02˜κ03˜κ01000˜κ02000˜κ03000⎞⎟ ⎟ ⎟ ⎟⎠, (2.3c)

where (2.3a) is the most general Ansatz for a symmetric and traceless tensor constructed from two four-vectors. The second term on the right-hand side of (2.3a) vanishes for the special choice (2.3b).

With the replacement rules given in KlinkhamerRisse2008b (), we can express our parameters in terms of the Standard Model Extension (SME) parameters KosteleckyMewes2002 (); BaileyKostelecky2004 ():

 ˜κ01 = −(˜κo+)(23), (2.4a) ˜κ02 = −(˜κo+)(31), (2.4b) ˜κ03 = −(˜κo+)(12). (2.4c)

Hence, the case considered here includes only parity-violating coefficients.

This parity-odd case may be of relevance, since it might reflect the parity-odd low-energy effective photon sector of a quantum theory of spacetime. Besides five parameters of the birefringent sector of modified Maxwell theory, whose coefficients are already strongly bounded, there is only one alternative parity-odd Lorentz-violating theory for the photon sector, which is gauge-invariant and power-counting renormalizable: MCS theory Carroll-etal1990 (). However, the MCS parameters are bounded to lie below by CMB polarization measurements Kostelecky:2008ts ().

Since the bounds are not as strong for the parity-odd case of nonbirefringent modified Maxwell theory defined by Eq. (2.3), a physical understanding of this case is of importance.

ii.3 Coupling to matter: Parity-odd modified QED

Modified photons are coupled to matter by the minimal coupling procedure to standard (Lorentz-invariant) spin- Dirac particles with electric charge and mass . This results in a parity-odd deformation of QED Heitler1954 (); JauchRohrlich1976 (); Veltman1994 (), which is given by the action

 Sparity-oddmodQED[˜κ0m,e,M]=Sparity-oddmodMax[˜κ0m]+SDirac[e,M], (2.5)

for , 2, 3 and with the modified-Maxwell term (2.1)–(2.3) for the gauge field and the standard Dirac term for the spinor field ,

 S Dirac[e,M]=∫R4d4x¯¯¯¯ψ(x)[γμ(i∂μ−eAμ(x))−M]ψ(x). (2.6)

Equation (2.6) is to be understood with standard Dirac matrices corresponding to the Minkowski metric .

Iii Dispersion relations

The field equations ColladayKostelecky1998 (); KosteleckyMewes2002 (); BaileyKostelecky2004 () of modified Maxwell theory in momentum space,

 MμνAν=0,Mμν≡kλkλημν−kμkν−2κμρσνkρkσ, (3.1)

lead to the following dispersion relations Casana-etal2009 () for the two physical degrees of freedom of electromagnetic waves (labeled ):

 ω1(k) = ˜κ01k1+˜κ02k2+˜κ03k3+√|k|2+(˜κ01k1+˜κ02k2+˜κ03k3)2, (3.2a) ω2(k) = ˜κ01k1+˜κ02k2+˜κ03k3+√1+(˜κ01)2+(˜κ02)2+(˜κ03)2|k|, (3.2b)

for wave vector and with the terms linear in the components explicitly showing the parity violation. To first order in , the dispersion relations are equal for both modes, but they differ at higher order.222It is evident that the so-called nonbirefringent Ansatz (2.2) is only nonbirefringent to first order in . Nevertheless we will still use the term “nonbirefringent” in order to distinguish from the nine-dimensional parameter sector of modified Maxwell theory, which shows no birefringence at least to first-order Lorentz violation, from the remaining ten coefficients. In the latter parameter region birefringent modes emerge already at first order with respect to the Lorentz-violating parameters KosteleckyMewes2002 (). With the modified Coulomb and Ampère law it can be shown that the dispersion relations (3.2) indeed belong to physical photon modes. The procedure given in ColladayKostelecky1998 () eliminates dispersion relations of unphysical, i.e. scalar and longitudinal, modes from the field equations. The two are given by

 ω0(k)=ω3(k)=|k|, (3.3)

where the index “0” refers to the scalar and the index “3” to the longitudinal degree of freedom of the photon field.

The dispersion relations (3.2) can be cast in a more compact form by defining components of the wave-vector which are parallel or orthogonal to the background “three-vector” :

 (3.4)

where and . By doing so, it is possible to write the dispersion relations (3.2) as follows:

 ω1(k⊥,k∥)=Ek∥+√k2⊥+(1+E2)k2∥, (3.5a) ω2(k⊥,k∥)=Ek∥+√1+E2|k|, (3.5b) where the three Lorentz-violating parameters ˜κ01, ˜κ02, and ˜κ03 are contained in the single parameter E that is defined as E≡|ζ|≡√(˜κ01)2+(˜κ02)2+(˜κ03)2. (3.5c)

It is obvious that , whereas each single parameter , , and can be either positive or negative. From the first definition of Eq. (3.4) we see that negative parameters , , are mimicked by a negative .

The phase and group velocity Brillouin1960 () of the above two modes can be cast in the following form for small enough :

 vph,1≡ω1|k|=1+Ecosθ+E22cos2θ+O(E3), (3.6a) vph,2≡ω2|k|=1+Ecosθ+E22+O(E3), (3.6b)
 vgr,1≡∣∣∣∂ω1∂k∣∣∣=1+Ecosθ+E22+O(E3), (3.7a) vgr,2≡∣∣∣∂ω2∂k∣∣∣=1+Ecosθ+(1+sin2θ)E22+O(E3), (3.7b)

where is the angle between the three-momentum and the unit vector : .

To leading order in , the velocities above are equal:

 vph,1=vph,2=vgr,1=vgr,2. (3.8)

Furthermore, Eqs. (3.6), (3.7) show that both phase and group velocity can be larger than 1. However, what matters physically is the velocity of signal propagation, which corresponds to the front velocity Brillouin1960 ():

 vfr≡limk↦∞vph. (3.9)

Equation (3.9) can be interpreted as the velocity of the highest-frequency forerunners of a signal. As can be seen from Eq. (3.6), and hence also do not depend on the magnitude of the wave vector, but only on its direction. For , we obtain , where

 vfr < 1forπ/2<θ<3π/2, (3.10a) vfr ≥ 1for0≤θ≤π/2∨3π/2≤θ<2π. (3.10b)

Observe that, for small enough , having or does not depend on the Lorentz-violating parameters but only on the direction in which the classical wave propagates. For completeness, we also give the phase velocities for propagation parallel and orthogonal to :

 vph,∥,1=ω1(k⊥,k∥)k∥∣∣∣k⊥=0=Esgn(k∥)+√1+E2=vph,∥,2, (3.11a) (3.11b)

with the sign function

 sgn(x)=⎧⎪⎨⎪⎩1forx>0,0forx=0,−1forx<0. (3.12)

Note that the latter results are in agreement with the inequalities of Eq. (3.10). We conclude that the front velocity can be larger than 1 for the wave vector pointing in certain directions. That leads us to the issue of microcausality, which will be discussed in Sec. VII.

Iv Propagator in the Feynman gauge

So far, we have investigated the dispersion relations of the classical theory. For a further analysis, especially concerning the quantum theory, the gauge propagator will be needed. The propagator is the Green’s function of the free field equations (3.1) in momentum space. In order to compute it the gauge has to be fixed. We decide to use the Feynman gauge Veltman1994 (); ItzyksonZuber1980 (); PeskinSchroeder1995 (), which can be implemented by the gauge-fixing condition

 (4.1)

The following Ansatz for the propagator turns out to be useful:

 ˆGνλ∣∣Feynman=−i{ +ˆaηνλ+ˆbkνkλ+ˆcξνξλ+ˆd(kνξλ+ξνkλ) +ˆeζνζλ+ˆf(kνζλ+ζνkλ)+ˆg(ξνζλ+ζνξλ)}ˆK1. (4.2)

The propagator coefficients , , and the scalar propagator part follow from the system of equations with the differential operator

 (G−1)μν=ημν∂2−2κμϱσν∂ϱ∂σ, (4.3)

in Feynman gauge transformed to momentum space. Scalar products , , and will be kept in the result, in order to gain some insight in the covariant structure of the functions. However, we remark that, for the case considered, , , and .

Specifically, the propagator coefficients and the scalar propagators and , where appears in some of these coefficients, are given by

 ˆK1 = 22k⋅ξk⋅ζ+k2(2−ξ⋅ζ), (4.4a) ˆK2 ≡ 44k⋅ξk⋅ζ+ξ2(k⋅ζ)2+ζ2(k⋅ξ)2+k2(4−ξ2ζ2), (4.4b)
 ˆa=1, (4.5a) ˆb =−14k4{ΥˆK2−2χ(2k⋅ξk⋅ζ+k2(2−ξ⋅ζ))}, (4.5b) ˆc=14[k2ζ2−(k⋅ζ)2]ˆK2, (4.5c) ˆd=k⋅ξ(2(k⋅ζ)2−k2ζ2)+2k2k⋅ζ4k2ˆK2, (4.5d) ˆe=14[k2ξ2−(k⋅ξ)2]ˆK2, (4.5e) ˆf=k⋅ζ(2(k⋅ξ)2−k2ξ2)+2k2k⋅ξ4k2ˆK2, (4.5f) Υ ≡ −2k⋅ξk⋅ζ(2k2−k⋅ξk⋅ζ)+(k⋅ζ)2((k⋅ξ)2−k2ξ2) (4.5g) +(k⋅ξ)2[(k⋅ζ)2−k2ζ2]+k2[12k⋅ξk⋅ζ+ξ2(k⋅ζ)2 +ζ2(k⋅ξ)2+k2(4−ξ2ζ2)], ˆg=−14[2k2+k⋅ξk⋅ζ]ˆK2, (4.5h)

where definition (4.5g) enters (4.5b).

The poles of and can be identified with the dispersion relations obtained in Sec. III. From , that is

 2k⋅ξk⋅ζ+k2(2−ξ⋅ζ)∣∣k0=ω1=0, (4.6)

the dispersion relation (3.2a) of the mode is recovered. Similarly, the dispersion relation (3.2b) of the mode follows from , that is

 4k⋅ξk⋅ζ+ξ2(k⋅ζ)2+ζ2(k⋅ξ)2+k2(4−ξ2ζ2)∣∣k0=ω2=0, (4.7)

The third pole corresponds to the dispersion relation of scalar and longitudinal modes. This is clear from the fact that this pole appears only in the gauge-dependent coefficients , , and . These are multiplied by at least one photon four-momentum and vanish by the Ward identity,333assuming if they couple to a conserved current PeskinSchroeder1995 (). Since the Ward identity results from gauge invariance, it also holds for modified Maxwell theory, which is expected to be free of anomalies ColladayKostelecky1998 (). Because of parity violation the physical poles are asymmetric with respect to the imaginary -axis.

The above result (IV)– (4.5) equals the propagator given in Casana-etal2010 (). Every propagator coefficient, which contains the scalar propagator , is also multiplied by . Hence, both modes appear together throughout the propagator and the question arises, whether they can be separated. It can be shown that the propagator can also be written in the following form:

 ˆGμν(k)∣∣Feynman=∑n=1,2Ξ(n)μν(k0,k)(−iˆG(n)(k)), (4.8)

where the tensor structure is the same for both parts, hence

 Ξ(1)νλ=Ξ(2)νλ= +ˆaηνλ+ˆbkνkλ+ˆcξνξλ+ˆd(kνξλ+ξνkλ) +ˆeζνζλ+ˆf(kνζλ+ζνkλ)+ˆg(ξνζλ+ζνξλ), (4.9)

with the coefficients , …, from Eq. (4.5). The scalar propagator functions are then given by:

 ˆG(1)(k)=4ˆK1ˆK−12[(k⋅ξ)2−k2]ζ2+(k⋅ζ)2,ˆG(2)(k)=−4[(k⋅ξ)2−k2]ζ2+(k⋅ζ)2. (4.10)

The first part contains both polarization modes encoded in and , whereas the second part does not involve any mode. The denominator that appears in both parts does not have a zero with respect to , hence it contains no dispersion relation. So it does not seem that the polarization modes can be separated, such that each propagator part contains exactly one of the modes.

Finally, we can state that the structure of the propagator of parity-odd nonbirefringent modified Maxwell theory is rather unusual. In the next section we will compute the polarization vectors.

V Polarization vectors

In what follows, the physical (transverse) degrees of freedom will be labeled with (1) and (2), respectively. For a fixed nonzero “three-vector” and a generic wave vector , the polarization vector of the mode reads

 (ε(1)μ)=1√N′(0,ζ×k)/∣∣ζ×k∣∣, (5.1)

where is a normalization factor to be given later. The polarization vector of the mode is orthogonal to (5.1) and has a longitudinal component. It is given by

 (ε(2)μ)=1√N′′1√|ε(2)|2−(ε0)2(ε0,ε(2)), (5.2)

with

 ε0=14(k2−(k⋅ξ)2)((k⋅ζ)2−ζ2[k2−(k⋅ξ)2]), (5.3a) ε(2) =(2|k|2|ζ|2−|k×(k×ζ)|2|k|2+2√1+|ζ|2|k|(k⋅ζ))k×(k×ζ) +(√1+|ζ|2|k|+k⋅ζ)|k×(k×ζ)|2|k|2k. (5.3b)

The polarization vector is a solution of the field equations (3.1), when is replaced by from Eq. (3.2a). The polarization is the corresponding solution for replaced by from Eq. (3.2b). The normalization factors in (5.1) and in (5.2) can be computed from the 00–component of the energy-momentum tensor. Note that the above polarization vectors have been calculated in the Lorentz gauge, .

For the Lorentz-violating decay processes considered, both the and the polarization modes contribute.

 ¯¯¯ε(1)με(1)ν =1N′{−ημν+ˆΓ1kμkν+ˆΔ1(kμξν+ξμkν)+ˆΛ1(kμζν+ζμkν) +ˆΦ1ξμξν+ˆΨ1ζμζν+ˆΘ1(ξμζν+ζνξμ)}∣∣k0=ω1, (5.4)

with

 ˆΓ1=ζ2Q,ˆΔ=ζ2[2k2+k⋅ξk⋅ζ]Q(ζ⋅k),ˆΛ1=−k⋅ζQ, (5.5a) ˆΘ1=−2k2+k⋅ξk⋅ζQ,ˆΨ1=k2ξ2−(k⋅ξ)2Q,ˆΦ1=k2ζ2−(k⋅ζ)2Q, (5.5b) N′=1ω1√k2⊥+(1+E2)k2∥,Q=ζ2[k2−(k⋅ξ)2]−(k⋅ζ)2, (5.5c)

where is given by (3.5a). The denominator vanishes only for or . If the polarization tensor of the mode is contracted with a gauge-invariant expression using the Ward identity,444This means dropping terms that are proportional to at least one external four-momentum , which we denote by the word “truncated” it can be replaced by :

 ¯¯¯ε(1)με(1)ν↦Πμν|λ=1, (5.6a) Πμν|λ=1 ≡¯¯¯ε(1)με(1)ν∣∣truncated =1N′{−ημν+ˆΦ1ξμξν+ˆΨ1ζμζν+ˆΘ1(ξμζν+ζνξμ)}∣∣k0=ω1. (5.6b)

The polarization tensor of the mode is lengthy and is best written up in terms of and defined in (3.4).

 ¯¯¯ε(2)με(2)ν =1N′′{+ˆΓ2kμkν+ˆΔ2(kμξν+ξμkν)+ˆΛ2(kμζν+ζμkν) +ˆΦ2ξμξν+ˆΨ2ζμζν+ˆΘ2(ξμζν+ζμξν)}∣∣k0=ω2, (5.7)

with

 ˆΓ2=E4N[Ek∥k2⊥+(2k2∥+k2⊥)ω2]2, (5.8a) (5.8b) ˆΛ2=−E32N|k|2(2k∥ω2+Ek2⊥)[Ek∥k2⊥+(2k2∥+k2⊥)ω2], (5.8c) ˆΦ2=E4N[k2⊥(k2⊥−ω22)−Ek∥k2⊥ω2+k2∥(k2⊥−2ω22)]2, (5.8d) ˆΨ2=E24N|k|4(Ek2⊥+2k∥ω2)2, (5.8e) (5.8f) N′′=E4k2⊥2ω22N|k|2[ +(4k2∥+k2⊥)ω42+4Ek∥k2⊥ω32+(4k4∥+2k2⊥k2∥+(E2−2)k4⊥)ω22 (5.8g) where N=|ε(2)|2−(ε0)2, (5.8h)

and is given by (3.5b). Again, if the tensor is contracted with a gauge-invariant expression, it can be replaced by :

 ¯¯¯ε(2)με(2)ν↦Πμν|λ=2, (5.9a) Πμν|λ=2 ≡¯¯¯ε(2)με(2)ν∣∣truncated =1N′′{ˆΦ2ξμξν+ˆΨ2ζμζν+ˆΘ2(ξμζν+ζμξν)}∣∣k0=ω2. (5.9b)

Finally it holds that

 kμ(¯¯¯ε(1)με(1)ν)(k)=0,limE↦0kμ(¯¯¯ε(2)με(2)ν)(k)=0, (5.10)

where the second contraction only vanishes for due to the longitudinal part of .

The polarization vector (5.2) is normalized to unit length by . This normalization factor cancels in . Note that the metric tensor does not appear on the right-hand side of (5.9), whereas it does on the right-hand side of Eq.  (5.6).

Furthermore, note that each truncated polarization tensor and can be written in a covariant form. This behavior is different from the polarization vectors of standard QED,555Also in the isotropic and the parity-even anisotropic sector of modified Maxwell theory the polarization tensor of one single transversal mode cannot be decomposed covariantly Klinkhamer:2010zs (). where only the whole polarization sum is covariant.

It is now evident that not only is the structure of the photon propagator uncommon, but the polarization vectors are unusual as well. In the next section we will analyze how both results are connected.

Vi The optical theorem and unitarity

In order to investigate unitarity, the simple test of reflection positivity used in Ref. Klinkhamer:2010zs () for the isotropic case of modified Maxwell theory cannot be adopted, because there are now essentially two different scalar propagators, namely and from (4.5). Hence, we could either examine reflection positivity of the full propagator or study the optical theorem for physical processes involving modified photons. As unitarity of the S–matrix results in the optical theorem and the latter is directly related to physical observables, we choose to proceed with the second approach.

The optical theorem will also show how the modified photon propagator in Sec. IV is linked to the photon polarizations from the previous section. The following computations will deal with the physical process that we already considered for isotropic modified Maxwell theory Klinkhamer:2010zs () in the context of unitarity: annihilation of a left-handed electron and a right-handed positron to a modified photon . The fermions are considered to be massless particles, which renders their helicity a physically well-defined state. Neglecting the axial anomaly, which is of higher order with respect to the electromagnetic coupling constant, the axial vector current is conserved: . This is the simplest tree-level process including a modified photon propagator. It has no threshold and is allowed for both photon modes. We assume a nonzero Lorentz-violating parameter . Furthermore, the four-momenta of the initial electron and positron are not expected to be collinear.

If the optical theorem holds, the imaginary part of the forward scattering amplitude is related to the cross section for the production of a modified photon from a left-handed electron and a right-handed positron:

 2Im(\includegraphics[]consistency−modmax−parity−oddv1fig1opt−theorem1−lhs.pdf)\lx@stackrel?=∫dΠ1∣∣\includegraphics[]consistency−modmax−parity−oddv1fig2opt−theorem1−rhs.pdf∣∣2. (6.1)

Herein, is the corresponding one-particle phase space element. By performing an integration over the four-momentum of the virtual photon, the forward scattering amplitude is given by

 Mˆ1 =∫d4k(2π)4δ(4)(k1+k2−k)e2¯¯¯u(k1)γλ\mathds1−γ52v(k2)¯¯¯v(k2)γν\mathds1−γ52u(k1) =∫d4k(2π)4×1ˆK−11+iϵ(+ηνλ+ˆbkνkλ+ˆcξνξλ+ˆd(kνξλ+ξνkλ) =∫d4k(2π)4×1ˆK−11+iϵ(+ˆeζνζλ+ˆf(kνζλ+ζνkλ)+ˆg(ξνζλ+ζνξλ)), (6.2)

with the propagator coefficients , …, from Eq. (4.5). Recall, that the physical poles have to be treated via Feynman’s -prescription. Hence, the denominator